13 is the second star number and the seventh Fibonacci number. As it is an odd-indexed Fibonacci number, it is a Markov number, appearing in solutions to the Markov Diophantine equation: (1, 5, 13), (1, 13, 34), (5, 13, 194), …

There are 13 Archimedean solids.

13 goes into 999,999 exactly 76,923 times, so vulgar fractions with 13 in the denominator have six digit repeating sequences in their decimal expansions. It is thus the smallest half period prime (sequence A097443 in OEIS)

13 is the largest number whose factorial is less than 10¹⁰. This means 13 is often the largest factorial a pocket calculator can represent without scientific notation.

At 13, the Mertens function sets a new low of -3, subseded later at 31 with a value of -4.

13 is the only positive integer that is the fourth root of the sum of the squares of two successive positive integers (119 and 120).

The 13th root is the most famous integer root calculation record, because 13 is the first prime number over 10 such as the last digit of a 13th integer power is the same as the last digit of its 13th root.

In base 10, the smallest prime with a composite sum of digits is 13.

13 is a repdigit in base 3 (111).

I am not a triskaidekaphobian, but there is logic in numbers which cannot be denied, patterns which shape the world around me, numbers coming to me in my dreams, while I drive, while I eat, which fascinate me, dictate my will, shape my reality.